CS 542 (Fall 2025) Written Assignment 1 Bayes’ Theorem and Naïve Bayes Classification

Due September 15, 2025

1. Bayes’ Theorem

   You get an email. You know that 90% of your email is legitimate (L) while 10% is spam (S).

   1. Assume the following probabilities:

      • The probability that an email contains the word “Bitcoin” (B) if it is spam is 96%.

      • The probability that an email contains the word “Bitcoin” if it is legitimate is 5%.

        What is the probability that the your new email is spam given that it contains the word “Bitcoin”? Show your work!

   2. Assume the following probabilities:

      • The probability that an email contains the word “Covid” (C) if it is spam is 50%.

      • The probability that an email contains the word “Covid” if it is legitimate is 12%.

        What is the probability that the your new email is legitimate given that it contains the word “Covid”? Show your work!

        
        

2. Naïve Bayes

The following problem is from the Jurafsky and Martin book, Exercise 4.2, reproduced below.

Given the following short movie reviews, each labeled with a genre, either comedy or action:

and a new document D: fast couple shoot fly

compute the most likely class for D. Assume a naive Bayes classifier and use add-1 smoothing for the likelihoods.

Show your work! In particular, show all of the probability distributions involved in the model (namely, P (class) and P (feature|class)) and all of the steps used to calculate them. Create (conditional) probability tables such as those shown below.

Perform Laplace Smoothing to account for words that do not appear in one class.

Submission Instructions

Please submit your solutions (in PDF format - printed and scanned images are OK) to the drop box on Canvas.